Exactly solvable deformations of the oscillator and Coulomb systems and their generalization
Angel Ballesteros, Alberto Enciso, Francisco J. Herranz, Orlando Ragnisco, Danilo Riglioni
EExactly solvable deformations of the oscillator and Coulombsystems and their generalization ´Angel Ballesteros a , Alberto Enciso b , Francisco J. Herranz a, Orlando Ragnisco c and Danilo Riglioni da Departamento de F´ısica, Universidad de Burgos, E-09001 Burgos, Spain b Instituto de Ciencias Matem´aticas, CSIC, Nicol´as Cabrera 13-15, E-28049 Madrid, Spain c Dipartimento di Matematica e Fisica, Universit`a di Roma Tre and Istituto Nazionale di FisicaNucleare sezione di Roma Tre, Via Vasca Navale 84, I-00146 Roma, Italy d Centre de Recherches Math´ematiques, Universit´e de Montr´eal, H3T 1J4 2920 Chemin de latour, Montreal, CanadaE-mail: [email protected], [email protected], [email protected],[email protected], [email protected]
Abstract
We present two maximally superintegrable Hamiltonian systems H λ and H η that are defined,respectively, on an N -dimensional spherically symmetric generalization of the Darboux sur-face of type III and on an N -dimensional Taub–NUT space. Afterwards, we show thatthe quantization of H λ and H η leads, respectively, to exactly solvable deformations (withparameters λ and η ) of the two basic quantum mechanical systems: the harmonic oscillatorand the Coulomb problem. In both cases the quantization is performed in such a way thatthe maximal superintegrability of the classical Hamiltonian is fully preserved. In particular,we prove that this strong condition is fulfilled by applying the so-called conformal Laplace–Beltrami quantization prescription, where the conformal Laplacian operator contains theusual Laplace–Beltrami operator on the underlying manifold plus a term proportional to itsscalar curvature (which in both cases has non-constant value). In this way, the eigenvalueproblems for the quantum counterparts of H λ and H η can be rigorously solved, and it isfound that their discrete spectrum is just a smooth deformation (in terms of the parameters λ and η ) of the oscillator and Coulomb spectrum, respectively. Moreover, it turns out thatthe maximal degeneracy of both systems is preserved under deformation. Finally, new fur-ther multiparametric generalizations of both systems that preserve their superintegrabilityare envisaged. Based on the contribution presented at “The 30th International Colloquium on Group Theoretical Methodsin Physics”, July 14–18, 2014, Ghent, Belgium. To appear in
Journal of Physics: Conference Series . a r X i v : . [ qu a n t - ph ] D ec Introduction
It is well known that if we consider a natural classical Hamiltonian system on the N -dimensional( N D) Euclidean space H = T ( p ) + U ( q ) , (1)the harmonic oscillator potential U ( q ) = ω q and the Coulomb potential U ( q ) = − k/ | q | definetwo maximally superintegrable (MS) systems (in the Liouville sense), since both systems areendowed with (2 N −
1) functionally independent and globally defined integrals of the motion. Inthe first case such integrals are provided by the components of the Demkov–Fradkin tensor [1, 2],and in the second one by the angular momenta together with the N components of the Runge–Lenz vector (see e.g. [3] and references therein). At the classical dynamical level, the footprintof superintegrability consists in the fact that all bounded trajectories of these two systems areclosed ones, a fact which is diretly related with Bertrand’s theorem [4]. Moreover, when thequantization of these systems is performed it is found that such superintegrability implies thattheir spectrum exhibits maximal degeneracy due to a superabundance of quantum integrals ofthe motion.In this paper we review two spherically symmetric deformations of the oscillator and Coulombsystems that define two new MS systems [5, 6]. As a consequence, their quantization [7, 8, 9]is shown to present maximal degeneracy in the spectra. At a first sight, the existence of suchdeformations could seem impossible since the only spherically symmetric potentials on the Eu-clidean space that are MS are just the oscillator and the Coulomb ones. Therefore, the additionof any radial perturbation on these systems leads to superintegrability breaking and thus to alack of maximal degeneracy in the spectra, a fact that is very well known in quantum pertur-bation theory. However, as we shall see, such superintegrable perturbations can be obtainedif both the potential and the kinetic energy are simultaneously deformed in a very precise way.Explicitly, the Hamiltonian (1) will be smoothly deformed into H µ ( q , p ) = T µ ( q , p ) + U µ ( q ) , (2)where µ can be regarded as a (generic) deformation parameter in such a manner that we willbe no longer working on the flat Euclidean space, but on a suitable curved space with metricand kinetic energy given byd s µ = N (cid:88) i,j =1 g ij ( q )d q i d q j , T µ ( q , p ) = 12 N (cid:88) i,j =1 g ij ( q ) p i p j . This fact will provide additional interesting geometric features to the systems we will deal with.In particular, we will see that the curved/deformed generalization of the Demkov–Fradkin tensorand of the Runge–Lenz vector do exist, and will be the essential tool to prove the MS propertyof the deformed systems.We recall that the quantization problem on curved spaces is clearly a non-trivial one, sincethe kinetic energy term T µ ( q , p ) is a function of both positions and momenta that createssevere ordering ambiguities. Nevertheless, we shall explicitly show that a quantization of H µ (2) that preserves the MS property is achieved through the conformal Laplacian quantization H c ,µ = ˆ T c ,µ + U µ = − (cid:126) c ,µ + U µ = − (cid:126) (cid:18) ∆ LB ,µ − ( N − N − R µ (cid:19) + U µ , (3)where R µ is the scalar curvature on the underlying N D curved manifold M µ , the operator ∆ c ,µ is the conformal Laplacian [14] and ∆ LB ,µ is the usual Laplace–Beltrami operator on M µ , i.e. ,∆ LB ,µ = N (cid:88) i,j =1 √ g ∂ i √ gg ij ∂ j , where g ij is the inverse of the metric tensor g ij and g is the corresponding determinant. Thelimit µ → s = d q sincelim µ → R µ = 0 , lim µ → ∆ c ,µ = lim µ → ∆ LB ,µ = ∆ = ∇ , lim µ → ˆ H c ,µ = − (cid:126) ∇ + U . We also recall that the quantization (3) can be related through a similarity transformation tothe Hamiltonian obtained by means of the so-called direct Schr¨odinger quantization prescriptionon conformally flat spaces [7, 8]ˆ H µ = ˆ T µ + U µ = − (cid:126) f µ ( r ) ∆ + U µ , where f µ ( r ) = f µ ( | q | ) is the conformal factor of the metric on M µ written as d s µ = f µ ( r ) d q .In this case, the scalar curvature reads R µ = − ( N − (cid:32) ( N − f (cid:48) µ ( r ) + f µ ( r ) (cid:0) f (cid:48)(cid:48) µ ( r ) + 2( N − r − f (cid:48) µ ( r ) (cid:1) f µ ( r ) (cid:33) . (4)In the next two sections, we review the exactly solvable deformations of the N D isotropic os-cillator ˆ H c ,λ [8] and the Coulomb system ˆ H c ,η [9], correspondingly. New results are sketched inthe last section by presenting the only possible multiparametric spherically symmetric general-izations of the above systems which are MS with quadratic integrals of motion, that is, the mostgeneric deformations that can be endowed, respectively, with a generalized Demkov–Fradkintensor and with a Runge–Lenz N -vector. The N D classical Hamiltonian system given by H λ ( q , p ) = T λ ( q , p ) + U λ ( q ) = p λ q ) + ω q λ q ) , (5)where λ and ω are real parameters and q , p ∈ R N are canonical coordinates and momenta, wasproven in [5] to be MS. The kinetic energy T λ ( q , p ) can be interpreted as the one generating3he geodesic motion of a particle with unit mass on a conformally flat space with metric and(non-constant) scalar curvature (4) given byd s λ = (1 + λ q )d q , R λ ( q ) = − λ ( N − (cid:0) N + 3 λ ( N − q (cid:1) (1 + λ q ) . Such a curved space is, in fact, an N D spherically symmetric generalization M λ [15, 16] of theDarboux surface of type III [17, 18, 19]. The limit λ → N D isotropic harmonic oscillator with frequency ω : H = 12 p + 12 ω q , d s = d q , R = 0 . The remarkable point is that H λ is a MS Hamiltonian, a fact that can be stated as follows. Proposition 1. [5, 6] (i) The Hamiltonian H λ (5) is endowed with the following constants ofmotion ( m = 2 , . . . , N ): • (2 N − angular momentum integrals: C ( m ) = (cid:88) ≤ i Let ˆ H c ,λ be the quantum Hamiltonian given by ˆ H c ,λ = − (cid:126) LB ,λ + ω q λ q ) − (cid:126) λ ( N − (cid:18) N + 3 λ ( N − q λ q ) (cid:19) , with ∆ LB ,λ = 1(1 + λ q ) ∆ + λ ( N − λ q ) ( q · ∇ ) . (7)4 i) ˆ H c ,λ commutes with the (2 N − quantum angular momentum operators ( m = 2 , . . . , N )ˆ C ( m ) = (cid:88) ≤ i 2) i (cid:126) λ λ ˆ q ) (ˆ q i ˆ p j + ˆ q j ˆ p i ) + ( N − (cid:126) λ ˆ q i ˆ q j (1 + λ ˆ q ) (cid:18) − N − (cid:19) − ( N − (cid:126) λ λ ˆ q ) δ ij − λ ˆ q i ˆ q j ˆ H c ,λ (ˆ q , ˆ p ) + ω ˆ q i ˆ q j , with i, j = 1 , . . . , N and such that ˆ H c ,λ = (cid:80) Ni =1 ˆ I c ,λ,ii .(ii) Each of the three sets { ˆ H c ,λ , ˆ C ( m ) } , { ˆ H c ,λ , ˆ C ( m ) } ( m = 2 , . . . , N ) and { ˆ I c ,λ,ii } ( i = 1 , . . . , N )is formed by N algebraically independent commuting observables.(iii) The set { ˆ H c ,λ , ˆ C ( m ) , ˆ C ( m ) , ˆ I c ,λ,ii } for m = 2 , . . . , N with a fixed index i is formed by (2 N − algebraically independent observables.(iv) ˆ H c ,λ is formally self-adjoint on the space L ( M λ ) , associated with the underlying DarbouxIII space, defined by (cid:104) Ψ | Φ (cid:105) c ,λ = (cid:90) M λ Ψ( q ) Φ( q ) (1 + λ q ) N/ d q . The complete solution to the eigenvalue problem along with the corresponding eigenfunctionsfor the case of positive deformation parameter λ is summarized in the following statement. Theorem 3. [8] Let ˆ H c ,λ be the quantum Hamiltonian (7) with λ > . Then:(i) The continuous spectrum of ˆ H c ,λ is given by [ ω λ , ∞ ) . There are no embedded eigenvaluesand its singular spectrum is empty.(ii) ˆ H c ,λ has an infinite number of eigenvalues, all of which are contained in (0 , ω λ ) . Their onlyaccumulation point is ω λ which is the bottom of the continuous spectrum.(iii) All the discrete eigenvalues of ˆ H c ,λ are of the form E λ,n = − λ (cid:126) (cid:18) n + N (cid:19) + (cid:126) (cid:18) n + N (cid:19) (cid:115) (cid:126) λ (cid:18) n + N (cid:19) + ω , n ∈ N . (9) (iv) The eigenfunction Ψ c ,λ of ˆ H c ,λ with eigenvalue E λ,n is given by Ψ c ,λ ( q ) = (1 + λ q ) (2 − N ) / N (cid:89) i =1 exp {− β q i / } H n i ( βq i ) , β = (cid:114) Ω (cid:126) , where H n i are Hermite polynomials with n i ∈ N such that n + · · · + n N = n and the deformedfrequency Ω is defined by Ω = (cid:113) ω − λE λ,n . 10 15 20 25 30 n E n Figure 1: The discrete spectrum E λ,n (9) for 0 ≤ n ≤ N = 3, (cid:126) = ω = 1 and λ = { , . , . , . } starting from the upper dot (straight) line corresponding to the isotropicharmonic oscillator with λ = 0, that is, E ,n . In the same order, E λ, = { . , . , . , . } and E λ, ∞ = {∞ , , , . } .Moreover, the bound states of this system satisfy E λ, ∞ = lim n →∞ E λ,n = ω λ , lim n →∞ ( E λ,n +1 − E λ,n ) = 0 . The discrete spectrum (9) is depicted in figure 1 as a function of n for several values of λ . As itwas remarked in the introduction, the spectrum turns out to be maximally degenerate since itcan be described as a function of just one quantum number n ∈ N . By taking into account thedefinition of n , the number of degenerate states D ( E λ,n ) for a given energy level E λ,n correspondsto all the possible combinations of { n i ∈ N } obeying to the constraint (cid:80) Ni =1 n i = n , namely D ( E λ,n ) = ( n + N − n !( N − , which for N = 3 reduces to the well known expression D ( E λ,n ) = ( n + 2)( n + 1) / An exactly solvable deformation of the Coulomb system Now we consider the N D Hamiltonian system given by H η = T η ( q , p ) + U η ( q ) = | q | η + | q | ) p − kη + | q | , (10)where η and k are real parameters. The metric and scalar curvature of the underlying manifold M η turns out to bed s η = (cid:18) η | q | (cid:19) d q , R η = η ( N − 1) 4( N − r + 3 η ( N − r ( η + r ) , r = | q | = (cid:112) q . We remark that the system (10) is directly related to a reduction [20] of the geodesic motionon the Taub–NUT space [21, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31]. In fact, this system can beregarded as an η -deformation of the N D Euclidean Coulomb problem with coupling constant k , since the limit η → H = 12 p − k | q | , d s = d q , R = 0 . Remarkably enough, the Hamiltonian H η turns out to be a MS classical system, and thisresult can be summarized as follows. Proposition 4. [6] (i) The Hamiltonian H η (10) is endowed with the (2 N − angular mo-mentum integrals (6) and Poisson-commutes with the R η,i components ( i = 1 , . . . , N ) of theRunge–Lenz N -vector given by R η,i = N (cid:88) j =1 p j ( q j p i − q i p j ) + q i | q | ( η H η ( q , p ) + k ) . (ii) The set {H η , C ( m ) , C ( m ) , R η.i } with m = 2 , . . . , N and a fixed index i is formed by (2 N − functionally independent functions. We also recall that the classical system H η has been fully solved in [32]. Next the quantumcounterpart of (10) can be obtained by applying (3), and reads: Proposition 5. [9] (i) The quantum Hamiltonian ˆ H c ,η given by ˆ H c ,η = − (cid:126) LB ,η − kη + | q | + (cid:126) η ( N − 2) 4( N − | q | + 3 η ( N − | q | ( η + | q | ) , with ∆ LB ,η = | q | η + | q | ∆ − η ( N − | q | ( | q | + η ) ( q · ∇ ) , (11) commutes with the (2 N − quantum angular momentum operators (8) as well as with thefollowing N Runge–Lenz operators ( i = 1 , . . . , N ) : ˆ R c ,η,i = 12 N (cid:88) j =1 (cid:18) ˆ p j + i (cid:126) η ( N − q j η + | ˆ q | ) ˆ q (cid:19) (ˆ q j ˆ p i − ˆ q i ˆ p j )+ 12 N (cid:88) j =1 (ˆ q j ˆ p i − ˆ q i ˆ p j ) (cid:18) ˆ p j + i (cid:126) η ( N − q j η + | ˆ q | ) ˆ q (cid:19) + ˆ q i | ˆ q | (cid:16) η ˆ H c ,η ( q , p ) + k (cid:17) . ii) Each of the three sets { ˆ H c ,η , ˆ C ( m ) } , { ˆ H c ,η , ˆ C ( m ) } ( m = 2 , . . . , N ) and { ˆ R c ,η,i } ( i = 1 , . . . , N )is formed by N algebraically independent commuting operators.(iii) The set { ˆ H c ,η , ˆ C ( m ) , ˆ C ( m ) , ˆ R c ,η,i } for m = 2 , . . . , N with a fixed index i is formed by (2 N − algebraically independent operators.(iv) ˆ H c ,η is formally self-adjoint on the Hilbert space L ( M η ) with the scalar product (cid:104) Ψ | Φ (cid:105) c ,η = (cid:90) M η Ψ( q ) Φ( q ) (cid:18) η | q | (cid:19) N/ d q . For a positive value of the deformation parameter η , the complete solution of the eigenvalueproblem for this quantum mechanical deformed Coulomb problem is the following. Theorem 6. [9] Let ˆ H c ,η be the quantum Hamiltonian (11) with k > and η > . Then:(i) The continuous spectrum of ˆ H c ,η is given by [0 , ∞ ) . There are no embedded eigenvalues andthe singular spectrum is empty.(ii) ˆ H c ,η has an infinite number of eigenvalues E η,n,l , depending only on the sum ( n + l ) andaccumulating at .(iii) The eigenvalues E η,n,l of ˆ H c ,η are of the form E η,n,l = − k (cid:126) (cid:0) n + l + N − (cid:1) + kη + (cid:113) (cid:126) (cid:0) n + l + N − (cid:1) + 2 (cid:126) kη (cid:0) n + l + N − (cid:1) , (12) such that the radial eigenfunction Φ c ,η ( r ) of ˆ H c ,η with eigenvalue E η,n,l reads Φ c ,η ( r ) = (cid:16) ηr (cid:17) − N r l exp (cid:32) − Kr (cid:126) (cid:0) n + l + N − (cid:1) (cid:33) L l + N − n (cid:32) Kr (cid:126) (cid:0) n + l + N − (cid:1) (cid:33) , where L αn are generalized Laguerre polynomials and the deformed coupling constant K reads K = k + η E η,n,l . Since ˆ H c ,η is a Hamiltonian with radial symmetry, its complete eigenfunction is so given byΨ c ,η = Φ c ,η ( r ) Y l ( θ ) where Y l ( θ ) denotes the usual hyperspherical harmonics, θ = ( θ , . . . , θ N − )and l is a vector of N − l = ( l , . . . , l N − , l N = l ) such that (see (8))ˆ C ( m ) Y l ( θ ) = (cid:126) l m ( l m + m − Y l ( θ ) , l m − l m − ≥ , m = 2 , . . . N. Notice also that the bound states of this system satisfylim n,l →∞ E η,n,l = 0 , lim n →∞ ( E η, n +1 − E η, n ) = 0 , n = n + l. As expected, the limit η → E η,n,l provides the well known formula for the standard Coulombeigenvalues E ,n,l E ,n,l = − k (cid:126) (cid:0) n + l + N − (cid:1) . n (cid:43) l (cid:43) (cid:72) N (cid:45) (cid:76)(cid:144) (cid:45) (cid:45) (cid:45) (cid:45) (cid:45) E n,l Figure 2: Discrete spectrum (12) for the fundamental and the three first excited states of theHamiltonian ˆ H c ,η (11) when η = { , . , . , . , } with (cid:126) = k = 1 and N ≥ 3. Note that theeffect of the η deformation is quite strong for the fundamental state, since it comes from theshift r → r + η in the usual Coulomb potential.And we find that the perturbative series for the eigenvalues of the deformed system ˆ H c ,η (11)reads E η,n,l = E ,n,l + η k (cid:126) (cid:0) n + l + N − (cid:1) − η k (cid:126) (cid:0) n + l + N − (cid:1) + O ( η ) . In figure 2 the eigenvalues of the fundamental and of the first three excited states are plottedfor different values of the deformation parameter η .As we can see from (12) the spectrum is maximally degenerate as, again, it depends on aunique principal quantum number n = n + l . The degeneracy D ( E η, n ) of a given energy level E η, n can be computed straightforwardly by taking into account that the cardinality D ( L l ) givenby the set of the hyperspherical harmonics { Y l ( θ ) } having the same quantum number l andsuch that ˆ C ( N ) { Y l ( θ ) } = (cid:126) l ( l + N − { Y l ( θ ) } reads [33] D ( L l ) = (2 l + N − l + N − l !( N − . From it we obtain that D ( E η, n ) = n (cid:88) l =0 D ( L l ) = (2 n + N − n + N − n !( N − . In particular, for N = 3 we obtain D ( E η, n ) = ( n + 1) , which coincides with the degeneracy ofthe energy levels of the undeformed Coulomb problem.9 Generalization So far we have reviewed some specific exactly solvable deformations of the oscillator andCoulomb potentials, which can be regarded as the most natural MS deformations beyond con-stant curvature. Nevertheless, there are more possible generalizations within this frameworkthat preserves the classical MS property and that would lead to other exactly solvable deformedoscillator and Coulomb systems. These arise within the classification of Bertrand Hamiltoniansformerly introduced in [34] and further developed in [35, 36, 37]. Such systems are MS andtheir underlying Bertrand spaces are spherically symmetric ones. If we require to keep quadratic integrals of motion, so generalizing the Demkov–Fradkin tensor and the Runge–Lenz N -vector,it can be shown that there only exists one possible generalization of the deformations of theoscillator and Coulomb systems here studied that depends on two deformation parameters.In particular, the two-parameter MS deformation of the oscillator system turns out to be H λ,ξ ( q , p ) = T λ,ξ ( q , p ) + U λ,ξ ( q ) = (1 − ξ q ) p λ q + ξ q ) + ω q λ q + ξ q ) , where ξ is a real parameter. Obviously, the limit ξ → H λ (5)The underlying manifold M λ,ξ is endowed with a conformally flat metric given byd s λ,ξ = (1 + λ q + ξ q )(1 − ξ q ) d q . And the corresponding scalar curvature (4) reads R λ,ξ ( r ) = − ( N − λr + ξr ) (cid:26) N (cid:0) λr + 6 ξr + λξr (cid:1)(cid:0) λ + 3 λξr + 2 ξr (3 + ξr ) (cid:1) − r ( λ − ξ )(1 − ξr ) (cid:27) , where recall that r = | q | = (cid:112) q .As far as the Coulomb system is concerned, the resulting two-parameter MS deformation isgiven by H η,ζ = T η,ζ ( q , p ) + U η,ζ ( q ) = (1 − ζ q ) | q | η + | q | + ηζ q ) p − k (1 + ζ q )( η + | q | + ηζ q ) , which generalizes the one-parameter Hamiltonian H η (10). Hence the metric of the underlyingspherically symmetric space M η,ζ and its scalar curvature are found to bed s η,ζ = ( η + | q | + ηζ q )(1 − ζ q ) | q | d q ,R η,ζ ( r ) = − ( N − r ( η + r + ηζr ) (cid:26) η (1 − ζr ) (cid:18) η + r (cid:0) ζr (6 η + r [2 + ηζr ]) (cid:1)(cid:19) − N (cid:0) η + r (4 + ηζr [6 − ζr ]) (cid:1)(cid:0) η − ζr (6 η + r [4 + 3 ηζr ]) (cid:1)(cid:27) . 10t is worth stressing that M η,ζ turns out to be the N D spherically symmetric generalization ofthe Darboux surface of type IV [17, 18, 19] constructed in [15, 16].Consequently, by applying the conformal Laplacian quantization (3) to the above two-parameter Hamiltonians, new exactly solvable systems, ˆ H c ,λ,ξ and ˆ H c ,η,ζ , would be obtained asdeformations of the oscillator and Coulomb systems. Their solution would generalize the resultspresented in theorems 3 and 6. Work on this line is currently in progress. Acknowledgments This work was partially supported by the Spanish MINECO through the Ram´on y Cajal pro-gram (A.E.) and under grants MTM2013-43820-P (A.B and F.J.H.) and FIS2011-22566 (A.E.),by the Spanish Junta de Castilla y Le´on under grant BU278U14 (A.B., A.E. and F.J.H.), bythe ICMAT Severo Ochoa under grant SEV-2011-0087 (A.E.), and by a postdoctoral fellowshipfrom the Laboratory of Mathematical Physics of the CRM, Universit´e de Montr´eal (D.R.). References [1] Demkov Y N 1959 Soviet Phys. JETP Amer. J. Phys. Commun. Math. Phys. C.R. Acad. Sci. Paris Physica D SIGMA Phys. Lett. A Ann. Phys. Ann. Phys. General Relativity (Chicago: The University of Chicago Press)[11] Liu Z J and Qian M 1992 Trans. Amer. Math. Soc. Mathematical Topics Between Classical and Quantum Mechanics (NewYork: Springer)[13] Michel J P, Radoux F and Silhan J 2014 SIGMA Geom. Funct. Anal. Phys. Lett. B Ann. Phys. Le¸cons sur la th`eorie g`en`erale des surfaces vol 4 ed G Darboux (New York:Chelsea) p 368[18] Kalnins E G, Kress J M, Miller W Jr and Winternitz P 2003 J. Math. Phys. Phys. Part. Nuclei J. Phys. A: Math. Gen. Phys. Lett. B Phys. Lett. A Nucl. Phys. B Phys. Lett. B Comm. Math. Phys. J. Math. Phys. J. Math. Phys. Class. Quantum Grav. Class. Quantum Grav. J. Geom. Phys. Class. Quantum Grav. preprint arXiv:1411.3571[33] Atkinson K and Han W 2012 Spherical Harmonics and Approximations on the Unit Sphere:An Introduction Lecture Notes in Mathematics (New York: Springer) p 16[34] Perlick V 1992 Class. Quantum Grav. Class. Quantum Grav. J. Phys. A: Math. Theor. preprintpreprint